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It is noteworthy that this equation describes the gradient flow of the Dirichlet-IntegralE(u):=12∫ℝ|u′|2dx,
E(u) := \frac{1}{2}\int_\mathbb{R} |u'|^2 \; d x
\,,
i.e. the fastest way to descrease this functional in the L2L^2-norm.

This means that the heat equation has the effect of regularizing a bumpy function into one that varies less and less.

Another remarkable aspect is that the familiar solutions (now expressed more generally for the Laplace operator in nn-dimensions)
u(x,t)=1(2πt)n/2exp(−|x|4t)
u(x,t) = \frac{1}{(2\pi t)^{n/2}} \exp \left( - \frac{|x|}{4t} \right)
are self-similar in that for any λ>0\lambda \gt 0 we have
u(x,t)=u(λx,λ2t).
u(x,t) = u(\lambda x, \lambda^2 t)
\,.

This says that the mm-spatial derivative DmuD^m u of uu is bounded, over a ball of radius R/2R/2 by the given prefactor time the supremum of uu itself over time and over a ball of radius RR.

Furthermore, there is another way to look at the regularization property of the heat equation, namely by realizing it as the gradient flow of the entropy functional∫ulogudx
\int u \mathrm{log}u \; d x
but now with respect to the Wasserstein metric (this applies to u>0u \gt 0).

There is an estimate, called the Li-Yau Harnack estimate, which says that
Δu≥|Dlogu|2−n2t.
\Delta u \geq |D \mathrm{log}u|^2 - \frac{n}{2t}
\,.
This holds as stated for ℝn\mathbb{R}^n and with slight modifications for arbitrary Riemannian manifolds whose curvature is bounded from below.

Estimates of this sort play an important role for understanding the following theory.

The regularization property of the heat equation has an analog in the following equation that describes curve shortening:

Consider any closed curve
Γ(⋅,0):S1→ℝ2
\Gamma(\cdot,0) : S^1 \to \mathbb{R}^2
in the real plane and let
t↦Γ(⋅,t)t \mapsto \Gamma(\cdot,t) be a 1-parameter family of such curves satisfying the equation
ddtΓ(p,t)=κ⇀(p,t)=κ(p,t)ν⇀(p,t),
\frac{d}{dt}
\Gamma(p,t)
=
\vec \kappa(p,t)
=
\kappa(p,t)\vec \nu(p,t)
\,,
where κ⇀\vec\kappa is the second arc-length derivative of Γ\Gammaκ⇀(p,t)=κ(p,t)ν⇀(p,t)=ddst2Γ(p,t),
\vec \kappa(p,t) = \kappa(p,t) \vec \nu(p,t)
=
\frac{d}{d s_t^2} \Gamma(p,t)
\,,
ν⇀(p,t)\vec \nu(p,t) is the unit normal vector to the curve at (p,t)(p,t) and κ(p,t)\kappa(p,t) is the spped of the curve at that point.

This is a quasi-linear differential equation (quasi since the arc-length depends on time). It describes again a gradient flow (with respect to the L2L^2-norm), now simply of the length of the curve
E(Γ)=∫S1ds.
E(\Gamma) = \int_{S^1} d s
\,.

It is a fact that under this flow, an embedded curve remains embedded. Meaning that a curve which doesn’t intersect itself to start with will never intersect itself in the future.

As an example, consider a curve which is initially a circle of radius R0R_0 centered at x0x_0 in ℝ2\mathbb{R}^2,
Γ0(S1)=SR01(x0).
\Gamma_0(S^1) = S^1_{R_0}(x_0)
\,.
Then under the above flow it will remain circular and centered at x0x_0Γt(S1)=SRt1(x0)
\Gamma_t(S^1) = S^1_{R_t}(x_0)
but shrink in radius according to
Rt=R02−2t.
R_t = \sqrt{R_0^2 - 2t}
\,.
This goes on until
t=T:=R02/2
t = T := R_0^2/2
at which point the curve has collapes to a (“round”) point and the flow equation diverges.

The interesting thing is that any embedded curve inside such a circular curve will also shrink, and will never be overtaken by the outer circular curve – hence will also shrink to a point – but always to a “round” point, i.e. no matter how wiggly it was to start with, it will always completely unwind to a nice circular curve just before collapsing to a point.

Theorem (Grayson, Gage-Hamilton): If Γ0(S1)\Gamma_0(S^1) is an embedded curve then Γt(S1)\Gamma_t(S^1) contracts smoothly to a (“round”) point in finite time.

The idea of the proof is this: one analyzes all possible ways that the extrinsic curvature
|κ⇀|→∞
|\vec \kappa| \to \infty
can become singular for t→Tt \to T. One uses the fact that all self-similar solutions are exact circles as in the above example and concludes that hence all singularities must be of this shape, too.

Now we go to higher dimensions, but still consider embedded geometries.

For an nn-dimensional something embedded in ℝn+1\mathbb{R}^{n+1}F0:Mn→ℝn+1
F_0 : M^n \to \mathbb{R}^{n+1}
we define a flow by the quasi-linear parabolic system
ddtF(p,t)=H⇀(p,t)=(λ1+⋯+λn)(p,t)⋅ν⇀(p.t):=ΔtF(p,t).
\frac{d}{dt}F(p,t)
=
\vec H(p,t)
=
(\lambda_1 + \cdots + \lambda_n)(p,t) \cdot \vec\nu(p.t)
:=
\Delta_t F(p,t)
\,.
Here the λi\lambda_i are the principal extrinsic curvatures, i.e. the nn eigenvalues of the the second fundamental form of the hypersurface.

This flow is, once again, a gradient flow, now for the nn-dimensional “area”:
E(F)=∫Mndμ.
E(F) = \int_{M^n} d\mu
\,.

Again, one can study the shrinking solutions of this flow. Those of curvature of definite signs are the nn-spheres and the nn-cyclinder.

The intrisic analog of this is Hamilton’s Ricci-Flow (from 1982).
This is a a manifold with a family of metrics t↦g(t)t \mapsto g(t) that flow according to the equation
ddtgij=−rRij(g),
\frac{d}{dt}g_{ij} = -r R_{ij}(g)
\,,
where RijR_{ij} are the components of the Ricci curvature tensor.

We can see how this is close to the extrinsic setup considered above by noticing that for any choice of local coordianetes we have an expansion
Rij=Δgij+⋯.
R_{ij} = \Delta g_{ij} + \cdots
\,.
The Ricci tensor hence indeed plays the role of the Laplace operator, but now in a diffeomorphism invariant context.

This diffeomorphism invariance of the Ricci flow is one of its main beauties, but is also what makes handling it more subtle.

The other big problem is the understanding and handling of the singularities of this Ricci flow.

(The main insight by Perelman is, roughly (as far as I understood) that by adjoining the dilaton field to the gravitational (= metric) field one is able to handle a dynamical re-adjustment of diffeomorphism in such a way that the behaviour of the singularities is under better control. More on that in part II. -urs)

Posted at February 2, 2007 12:16 PM UTC

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Re: Huisken on Uniformization, I

A comment in mathoverflow relates Ricci flows to renormalization. Because the later seems to be a universal concept with connections to arithmetics (e.g. in arxiv articles by Connes,Marcolli), I’d like to know more about that. Do you know where one could read about such a connection between Ricci flow and renormalization?

Re: Huisken on Uniformization, I

Re: Huisken on Uniformization, I

Yes, Ricci flow is the special case of renormalization group flow for the Polyakov sigma-model 2-d QFT – usually interpreted as describing the wolrdvolume theory of the bosonic string propagating in a background gravitational field, encoded by the Riemannian metric. The background Riemannian metric is the collection of coupling constants for the worldvolume theory, and these “flow” under renormalization of the worldsheet theory. This is the Ricci flow.

This is mentioned a bit more explicitly in the followuo entry Huisken on renormalization II. In particular, there it is mentioned that Perelman’s method involves considering the more general situation where the string sees not only a gravitationa background field, but also the Kalb-Ramand and notably the dilaton background fields. These three fields are precisely the three massless bosonic background fields of the bosonic string.

And notice that this Ricci-flow problem for the bosonic string is effectively the source where much of the interest in string theory origininated in: the worldvolume theory of the string is required to be conformal, which means that it sits at a fixed point of the renormalization group/Ricci flow. Which in the presence of only the gravitational background field says hence that the background metric has to be Ricci flat. Which happens to be precisely the Einstein-equation for gravity in the absence of other fields. If the other fields are taken into account, the equation for the fixed point of the Ricci flow changes accordingly to the equation of gravity coupled to other fields.

So the Ricci flow equation is one of the indications that strings may know about gravity.

Re: Huisken on Uniformization, I

Many thanks! An other expert’s answer as copy here. Amazing how things are interconnected, Manin’s castle seems to live in a very high dim space, as so many different parts of it are within a short distance :)